Drew realized that Michelle must be a student from his college. She had never met this Michelle and knew nothing about her, though. Impersonating her was going to be difficult, if not impossible.
Worse, Drew realized that she now had the distinct prospect of running in to someone who knew the real Michelle, and he had no idea how to react in such an instance.
Finally, Drew still had nowhere to go. She had no idea where Michelle lived, or if she lived alone. Despite the fact that she knew who she was, she was still alone, soaking wet, and had no place to find shelter.